Search in the Private tab always using the default SE even when the default SE changed to other SEs

[GeetaSarvadnya]

Description

Found while testing #26752

Search in the Private tab always using the default SE even when the default SE changed to other SEs

Steps to reproduce

Local is US - 1.45.127

1. Clean profile 1.45.127
2. Perform search in both NT and PT
3. Search is performed using `Brave` default SE as expected
4. Change default SE to NT=Bing PT=startpage
5. Repeat step 2
6. Search is performed using `Bing` SE in Normal tab as expected, but search is performed using `Brave` SE in PT
7. Change default SE to NT=DDG PT=Ecosia
8. Repeat step 2
9. Search is performed using DDG SE in Normal tab as expected, but search is performed using `Brave` SE in PT

locale is India - 1.45.127

1. Clean profile 1.45.127
2. Perform search in normal tab
3. Search is performed using `Gogle` SE as expected
4. Open an new NTP
5. Tap on the URL bar
6  Select default SE as `Bing` from the `select default search engine` modal popup
7. Confirmed that defalu SE is changed as `Bing` in both NT and PT
8. Perform search in both NT and PT
9. Search is performed using `Bing` SE in both tabs as expected 
10. Change default SE, NT=DDG PT=startpage
11. Perform search in both NT and PT
12. Search is performed using DDG SE in Normal tab as expected, but search is performed using Bing SE in PT

Actual result

Search in the Private tab always using the default SE even when the default SE changed to other SEs

Expected result

Search in the Private tab should use the SE which is selected

Issue reproduces how often

Easy

Version/Channel Information:

• Can you reproduce this issue with the current Play Store version? Yes
• Can you reproduce this issue with the current Play Store Beta version? Yes
• Can you reproduce this issue with the current Play Store Nightly version?Yes

Device details

• Install type (ARM, x86): All
• Device type (Phone, Tablet, Phablet):
• Android version: 12

Brave version

1.45.127

Website problems only

• Does the issue resolve itself when disabling Brave Shields? NA
• Does the issue resolve itself when disabling Brave Rewards?NA
• Is the issue reproducible on the latest version of Chrome? NA

Additional information

cc: @brave/qa-team @SergeyZhukovsky @anthonypkeane @samartnik

[GeetaSarvadnya]

Dup of #25821